3.109 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^2 (d+e x)} \, dx\)

Optimal. Leaf size=115 \[ -\frac{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

-(d*e*(2*d + 3*e*x)*Sqrt[d^2 - e^2*x^2])/2 - ((3*d + e*x)*(d^2 - e^2*x^2)^(3/2))
/(3*x) - (3*d^3*e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 + d^3*e*ArcTanh[Sqrt[d^2
- e^2*x^2]/d]

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Rubi [A]  time = 0.359344, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{2} d e (2 d+3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(5/2)/(x^2*(d + e*x)),x]

[Out]

-(d*e*(2*d + 3*e*x)*Sqrt[d^2 - e^2*x^2])/2 - ((3*d + e*x)*(d^2 - e^2*x^2)^(3/2))
/(3*x) - (3*d^3*e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 + d^3*e*ArcTanh[Sqrt[d^2
- e^2*x^2]/d]

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Rubi in Sympy [A]  time = 47.8671, size = 99, normalized size = 0.86 \[ - \frac{3 d^{3} e \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2} + d^{3} e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} - \frac{d e \left (4 d + 6 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{4} - \frac{\left (3 d + e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(5/2)/x**2/(e*x+d),x)

[Out]

-3*d**3*e*atan(e*x/sqrt(d**2 - e**2*x**2))/2 + d**3*e*atanh(sqrt(d**2 - e**2*x**
2)/d) - d*e*(4*d + 6*e*x)*sqrt(d**2 - e**2*x**2)/4 - (3*d + e*x)*(d**2 - e**2*x*
*2)**(3/2)/(3*x)

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Mathematica [A]  time = 0.151217, size = 114, normalized size = 0.99 \[ -d^3 e \log (x)+d^3 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-\frac{d^3}{x}-\frac{4 d^2 e}{3}-\frac{1}{2} d e^2 x+\frac{e^3 x^2}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(5/2)/(x^2*(d + e*x)),x]

[Out]

Sqrt[d^2 - e^2*x^2]*((-4*d^2*e)/3 - d^3/x - (d*e^2*x)/2 + (e^3*x^2)/3) - (3*d^3*
e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 - d^3*e*Log[x] + d^3*e*Log[d + Sqrt[d^2 -
e^2*x^2]]

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Maple [B]  time = 0.016, size = 380, normalized size = 3.3 \[ -{\frac{1}{{d}^{3}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{2}x}{{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{e}^{2}x}{4\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{15\,d{e}^{2}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{15\,{e}^{2}{d}^{3}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{5\,{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{2}x}{4\,d} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{3\,d{e}^{2}x}{8}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{3\,{e}^{2}{d}^{3}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{e}{5\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{e}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-e{d}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+{e{d}^{4}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(5/2)/x^2/(e*x+d),x)

[Out]

-1/d^3/x*(-e^2*x^2+d^2)^(7/2)-1/d^3*e^2*x*(-e^2*x^2+d^2)^(5/2)-5/4*e^2/d*x*(-e^2
*x^2+d^2)^(3/2)-15/8*d*e^2*x*(-e^2*x^2+d^2)^(1/2)-15/8*e^2*d^3/(e^2)^(1/2)*arcta
n((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/5*e/d^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(
5/2)+1/4*e^2/d*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x+3/8*e^2*d*(-(x+d/e)^2*e^2+
2*d*e*(x+d/e))^(1/2)*x+3/8*e^2*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*
e^2+2*d*e*(x+d/e))^(1/2))-1/5*e/d^2*(-e^2*x^2+d^2)^(5/2)-1/3*e*(-e^2*x^2+d^2)^(3
/2)-e*d^2*(-e^2*x^2+d^2)^(1/2)+e*d^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x
^2+d^2)^(1/2))/x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298381, size = 532, normalized size = 4.63 \[ -\frac{8 \, d e^{7} x^{7} - 12 \, d^{2} e^{6} x^{6} - 48 \, d^{3} e^{5} x^{5} + 12 \, d^{4} e^{4} x^{4} + 48 \, d^{5} e^{3} x^{3} + 48 \, d^{6} e^{2} x^{2} - 48 \, d^{8} - 18 \,{\left (d^{3} e^{5} x^{5} - 8 \, d^{5} e^{3} x^{3} + 8 \, d^{7} e x + 4 \,{\left (d^{4} e^{3} x^{3} - 2 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 6 \,{\left (d^{3} e^{5} x^{5} - 8 \, d^{5} e^{3} x^{3} + 8 \, d^{7} e x + 4 \,{\left (d^{4} e^{3} x^{3} - 2 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (2 \, e^{7} x^{7} - 3 \, d e^{6} x^{6} - 24 \, d^{2} e^{5} x^{5} + 18 \, d^{3} e^{4} x^{4} + 48 \, d^{4} e^{3} x^{3} + 24 \, d^{5} e^{2} x^{2} - 48 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{4} x^{5} - 8 \, d^{2} e^{2} x^{3} + 8 \, d^{4} x + 4 \,{\left (d e^{2} x^{3} - 2 \, d^{3} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x^2),x, algorithm="fricas")

[Out]

-1/6*(8*d*e^7*x^7 - 12*d^2*e^6*x^6 - 48*d^3*e^5*x^5 + 12*d^4*e^4*x^4 + 48*d^5*e^
3*x^3 + 48*d^6*e^2*x^2 - 48*d^8 - 18*(d^3*e^5*x^5 - 8*d^5*e^3*x^3 + 8*d^7*e*x +
4*(d^4*e^3*x^3 - 2*d^6*e*x)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d
^2))/(e*x)) + 6*(d^3*e^5*x^5 - 8*d^5*e^3*x^3 + 8*d^7*e*x + 4*(d^4*e^3*x^3 - 2*d^
6*e*x)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (2*e^7*x^7 - 3
*d*e^6*x^6 - 24*d^2*e^5*x^5 + 18*d^3*e^4*x^4 + 48*d^4*e^3*x^3 + 24*d^5*e^2*x^2 -
48*d^7)*sqrt(-e^2*x^2 + d^2))/(e^4*x^5 - 8*d^2*e^2*x^3 + 8*d^4*x + 4*(d*e^2*x^3
- 2*d^3*x)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 21.0631, size = 386, normalized size = 3.36 \[ d^{3} \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) - d^{2} e \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(5/2)/x**2/(e*x+d),x)

[Out]

d**3*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/
(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-d/(x*sqrt(1 - e**2*x*
*2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True)) - d**2*e
*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(
d**2/(e**2*x**2) - 1), Abs(d**2/(e**2*x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**
2*x**2) + 1)) + I*d*asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1), True)) -
d*e**2*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2
)) + I*e**2*x**3/(2*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**
2*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True)) + e**3*Piecewise((x
**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError